Combining Regression Models to Estimate Missing Subgroup Outcomes

Summary

The document asks how to estimate outcomes for US men and US women from three separate height datasets: people in the US, men worldwide, and women worldwide. Each dataset has its own linear regression, and some predictors may overlap. The proposed approach is to take a weighted average of predictions from two models, but the questioner does not know how to choose the weights.

No solution or empirical evidence is included, so the document does not establish that a weighted average is appropriate. The example instead raises a data-combination problem: estimates for intersections of groups are unavailable, and the populations represented by the observed datasets differ. Any method would depend on assumptions about subgroup effects, population composition, predictor relationships, and how the datasets overlap. The document leaves those assumptions and the estimation strategy unresolved.

Key ideas

• The question concerns estimating outcomes for subgroups that are absent from the observed datasets.
• Separate linear regressions are fitted to three populations with potentially overlapping predictors.
• A weighted average of model predictions is proposed, but no rule for selecting weights is supplied.
• The document provides no answer or evidence that establishes a valid combining method.

Tags

# How to combine regression models?


# How to combine regression models?












Say I have three data sets of size $n$ each:

$y_1$ = heights of people from the US only

$y_2$ = heights of men from the whole world

$y_3$ = heights of women from the whole world

And I build a linear model for each with factors $x_i$, $i = 1,..., k$:

$\hat{y}_{j} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon_{j}$

with $\epsilon$ having the usual properties for OLS. And I may use a factor $x_i$ in more than one regression. My question is: How could I combine the regressions such that I can obtain estimates for:

$y_{12}$ = height of men from the US only

$y_{13}$ = height of women from the US only

for which I do not have data

I thought of perhaps some sort of weighting:

$ \hat{y}_{12} = w_{1} \hat{y}_{1} + (1 - w_{1}) \hat{y}_{2}$

but then I wouldn't know what to use for $w_1$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.